# Questions tagged [cardinals]

This tag is for questions about cardinals and related topics such as cardinal arithmetics, regular cardinals and cofinality. Do not confuse with [large-cardinals] which is a technical concept about strong axioms of infinity.

2,352 questions
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### Problem of cardinal assignment

A weak cardinal assignment is any definite operation on sets $A\mapsto |A|$ which satisfies (C1) and (C3), and it is a strong cardinal assignment if it also satisfies (C2). The cardinal numbers (...
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### Set $X$ such that $2^c \leq |X|\leq |(\ell^\infty)^*|$ [duplicate]

I want to prove that $2^c\leq|(\ell^\infty)^*|$. My question is: Is there a set $X$ with $2^c\leq|X|$ such that $f:X\to(\ell^\infty)^*$ is an injective function or $g:(\ell^\infty)^*\to X$ is a ...
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### S is a possibly infinite set. Prove |S| < |P(S)|

Suppose S is a set. Do not assume that S is finite. How can one prove that |S|<|P(S)|? (P(S) is the power set of S). Would I say something how the power set is the subset of S so that it contains ...
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### Diagonal argument applied to computable numbers

Upon applying the Cantor diagonal argument to the enumerated list of all computable numbers, we produce a number not in it, but seems to be computable too, and that seems paradoxical. For clarity, ...
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### for any infinite set A, |A| >= |N| [duplicate]

I try to prove that the power of any infinite set must be equal/bigger then the power of the natural א0. I tried to show that for any infinite set there exists a subset of it, that its power is the ...
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### Two well orderings of an infinite cardinal agree on a large set

I've seen this question but I'm having trouble following the proof given. This is an exercise from Kunen: If $\kappa$ is an infinite cardinal and $\triangleleft$ is a well ordering on $\kappa$, then ...
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### $\sigma$-ideal of subsets of $2^\omega$

I do not understand why here on the page 148 $\cal M^*_{2, K}$ is a $\sigma-$ideal of subsets of $2^\omega$. I even do not know the weaker statement: why it is an ideal?
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### $X \subset \mathbb{R}_{>0}$ exists $C$ for every $\{x_1, …, x_n\} \subset X$ $\sum^{n}_{i=1} x_i<C$ Then $X$ is countable

Let $X \subset \mathbb{R}_{>0}$ such that exists $C$ that, for every $\{x_1, ..., x_n\} \subset X$ $$\sum^{n}_{i=1} x_i<C.$$ Prove that $X$ is countable My intuition goes that if $X$ is not ...
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### Cardinality of $A$ and the Natural Numbers [closed]

The question asks to show that the set $A$ and the set of natural numbers have the same cardinality. Where $$A=(1,2,3)\times\mathbb{N}$$ So I know I have to prove a bijection, but I am having ...
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### cardinality proof: prove that for any set $A, \ 2^{|A|} \neq |\mathbb{N}|$ [duplicate]

Prove: for any set $A, \ 2^{|A|} \neq \aleph _{0}$ as $\aleph_{0} = |\mathbb{N}|$. my attempt: Suppose by contradiction that there exist a set $A$ such that $2^{|A|} = \aleph_{0}$, which Implies ...
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### Decomposition of an $\aleph_1$ set into $\aleph_1$ sets $\aleph_1$ [closed]

Can we justify the claim Any set of cardinality $\aleph_1$ can be expressed as the disjoint union of $\aleph_1$ sets of cardinality $\aleph_1$ in a simple yet reasonably-correct way?
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### Question about the existence of sets

If there is a sequence of set $a_{i}$ and $i \in I \wedge |I|>|\mathbb{R}|$, is the set $A$ which $\forall i (i \in I \rightarrow a_{i}\in A)$ and the set $\bigcup \limits_{i \in \mathbb{R}} a_{i}$ ...
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### Are infinite indexes multiplicative?

For a group $G$ and its subgroup $H$, the index of $H$ relative to $G$, denoted by $[G:H]$, is the cardinality of the set $\{ gH \mid g \in G \}$. It is known that $|G| = [G:H]|H|$ even if all ...
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### Does it make sense to say that “there are more non-Abelian groups than Abelian groups”?

Firstly, does the family of all non-isomorphic non-Abelian groups have a well defined cardinality? How about the family of all non-isomorphic Abelian groups? If they are both defined, how do they ...
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### cardinality of sets intersections

Given that: $$| A \cap \mathbb{R}| = |\mathbb{R}|$$ $$| B \cap \mathbb{R}| = |\mathbb{N}|$$ Can we say that: $$|A| = |\mathbb{R}|$$ $$|B| = |\mathbb{N}|$$ ? If not, can we say something for ...
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### Cardinality of set of words [duplicate]

How can we prove that for the countable alphabet $A$ the cardinality of set of all finite words over this alphabet $A^*$ is equal to $\aleph_0$
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### Is the least inaccessible cardinal equivalent to the first aleph fixed point? [duplicate]

Let $I$ be the least / first inaccessible cardinal. As inaccessible cardinas are all aleph fixed points, and they are regular, so each inaccessible cardinal is an aleph fixed point after the previous ...
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### Cardinality of sets of functions $\mathbb{R}\to \mathbb{N}$ and $\mathbb{N}\to \mathbb{R}$. [closed]

Let $B^A$ denote the set of all functions $A \to B$. Prove that $\left|\mathbb{R}^\mathbb{N}\right|<\left|\mathbb{N}^\mathbb{R}\right|$.
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### Cardinality of infinite dimensional vector space

Assume that V is an infinite dimensional vector space. I know that if V is a vector space over a field F, then |V|=max{dimV,|F|}. So if we take V=$\mathbb{R}$ and F=$\mathbb{Q}$ then |V|>|F| and |V|=...
### Cardinality of $[\lambda]^\kappa$
Let $\kappa \leq \lambda$ cardinals with $\lambda$ infinite, and $[\lambda]^\kappa=\{Y\subseteq\lambda : ot(Y,\in)=\kappa\}$. I want to show that $[\lambda]^\kappa \asymp\ ^\kappa\lambda$. I've ...
### How to show $\operatorname{card}(\omega+1)=\omega$
Apparently $\operatorname{card}(\omega+1)=\omega$. This means that there is an order $<$ on $\omega+1$ such that there is an isomorphism of ordered set $f$, $(\omega+1,<) \cong (\omega,\in)$, ...